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By F. William Lawvere (auth.), Aurelio Carboni, Maria Cristina Pedicchio, Guiseppe Rosolini (eds.)

With one exception, those papers are unique and entirely refereed learn articles on numerous functions of class idea to Algebraic Topology, good judgment and computing device technology. The exception is an exceptional and long survey paper through Joyal/Street (80 pp) on a becoming topic: it provides an account of classical Tannaka duality in this sort of method as to be obtainable to the final mathematical reader, and to supply a key for access to newer advancements and quantum teams. No services in both illustration thought or classification thought is thought. subject matters equivalent to the Fourier cotransform, Tannaka duality for homogeneous areas, braided tensor different types, Yang-Baxter operators, Knot invariants and quantum teams are brought and reports. From the Contents: P.J. Freyd: Algebraically whole categories.- J.M.E. Hyland: First steps in artificial area theory.- G. Janelidze, W. Tholen: How algebraic is the change-of-base functor?.- A. Joyal, R. road: An creation to Tannaka duality and quantum groups.- A. Joyal, M. Tierney: robust stacks andclassifying spaces.- A. Kock: Algebras for the partial map classifier monad.- F.W. Lawvere: Intrinsic co-Heyting barriers and the Leibniz rule in convinced toposes.- S.H. Schanuel: damaging units have Euler attribute and dimension.-

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I t x R ~ Id(R). In this first approach of the problem, we limit our attention to the case of commutative unital rings and define the equality by [r = s] = Ann ( r - s) where "Ann" stands for annihilator. This gives a representation of the ring R in the usual sense that the global elements of ~ are in bijection with the elements of It. Moreover, given one of the classical constructions on a spectrum of R obtained via a quotient q: I d ( R ) ~ O ( S p ( R ) ) the composite R x It Id(it) O(Sp(R)) is just the corresponding classical sheaf representation of the ring.

5] C. MULVEY, Representations of rings and modules, Springer LNM 753, 1980, 542587. This paper is in final form and will not be published elsewhere. NORMALIZATION EQUIVALENCE, KERNEL EQUIVALENCE AND ~ CATEGORIES Dominique Bourn Fac. de MathEmatiques, Universit6 de Picardie 33 rue St Leu, 80039 Amiens France. In a recent paper [4], A. Carboni gave an interesting characterization of the categories of affine spaces, i. e. slices of additive categories, by means of a "modularity" condition, relating coproducts and puUbacks, which is a categorical version of the modularity condition for lattices, in the same way as the distributive categories are the categorical version of the distributive lattices.

Examole 1. When IE is modular, IE has split pullbacks and finite products. Moreover the fibration p is trivial because of the kernel equivalence. Finally, following the proposition 4, IE has split puschouts. So the modular categories are essentially affine. The previous terminology is due to the following result : Prot~osition 5. e. each fiber and each change of base functor is additive). PrQof. e. the fiber is pointed). Furthermore, IE having split pullbacks, each fiber admits finite products and each change of base functor preserves them.

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